zbMATH — the first resource for mathematics

Pointwise pinched manifolds are space forms. (English) Zbl 0587.53042
Geometric measure theory and the calculus of variations, Proc. Summer Inst., Arcata/Calif. 1984, Proc. Symp. Pure Math. 44, 307-328 (1986).
[For the entire collection see Zbl 0577.00014.]
Theorem 1. If the unnormalized Ricci flow \(\partial g/\partial t=-2\cdot ric\) starts at a metric with positive curvature operator, then the curvature operator stays positive. Corollary to theorem 2. If M has positive sectional curvatures which are pointwise \(\delta\) (n) pinched, then the normalized Ricci flow converges to a metric of constant sectional curvature \((\delta (n)=1-n^{-3/2}\) suffices for large n).
The estimates are based on splitting tensors with curvature tensor symmetries into their irreducible components and then on following Hamilton’s procedure with improved estimates.
Reviewer: H.Karcher

53C20 Global Riemannian geometry, including pinching
58J35 Heat and other parabolic equation methods for PDEs on manifolds